2,885 research outputs found
Entropy of Some Models of Sparse Random Graphs With Vertex-Names
Consider the setting of sparse graphs on N vertices, where the vertices have
distinct "names", which are strings of length O(log N) from a fixed finite
alphabet. For many natural probability models, the entropy grows as cN log N
for some model-dependent rate constant c. The mathematical content of this
paper is the (often easy) calculation of c for a variety of models, in
particular for various standard random graph models adapted to this setting.
Our broader purpose is to publicize this particular setting as a natural
setting for future theoretical study of data compression for graphs, and (more
speculatively) for discussion of unorganized versus organized complexity.Comment: 31 page
Local limit theorems via Landau-Kolmogorov inequalities
In this article, we prove new inequalities between some common probability
metrics. Using these inequalities, we obtain novel local limit theorems for the
magnetization in the Curie-Weiss model at high temperature, the number of
triangles and isolated vertices in Erd\H{o}s-R\'{e}nyi random graphs, as well
as the independence number in a geometric random graph. We also give upper
bounds on the rates of convergence for these local limit theorems and also for
some other probability metrics. Our proofs are based on the Landau-Kolmogorov
inequalities and new smoothing techniques.Comment: Published at http://dx.doi.org/10.3150/13-BEJ590 in the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Archimedes, Gauss, and Stein
We discuss a characterization of the centered Gaussian distribution which can
be read from results of Archimedes and Maxwell, and relate it to Charles
Stein's well-known characterization of the same distribution. These
characterizations fit into a more general framework involving the beta-gamma
algebra, which explains some other characterizations appearing in the Stein's
method literature.Comment: 13 pages, 2 figure
Approximating stationary distributions of fast mixing Glauber dynamics, with applications to exponential random graphs
We provide a general bound on the Wasserstein distance between two arbitrary
distributions of sequences of Bernoulli random variables. The bound is in terms
of a mixing quantity for the Glauber dynamics of one of the sequences, and a
simple expectation of the other. The result is applied to estimate, with
explicit error, expectations of functions of random vectors for some Ising
models and exponential random graphs in "high temperature" regimes.Comment: Ver3: 24 pages, major revision with new results; Ver2: updated
reference; Ver1: 19 pages, 1 figur
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